Electricity is consumed between the hours of midnight (t = 0) and 6 a.m. (t = 6). Selected values of the rate of consumption are shown in the table below, with t measured in hours past midnight, and R(t) measured in KW per hour. Use 4 trapezoids to estimate the total amount of electricity to the nearest KW consumed between midnight and 6 a.m.
t (hours) 0 1 2 5 6
R(t) KW/hr 400 200 350 200 150
To estimate the total amount of electricity consumed between midnight and 6 a.m. using four trapezoids, you can use the Trapezoidal Rule. The Trapezoidal Rule is a method for approximating the definite integral of a function by dividing the interval into smaller subintervals and approximating the area under the curve using trapezoids.
Here's how you can apply the Trapezoidal Rule to this problem:
1. First, divide the interval [0, 6] into four equal subintervals. In this case, each subinterval will have a width of (6 - 0)/4 = 1.5 hours. The subintervals are [0, 1.5], [1.5, 3], [3, 4.5], and [4.5, 6].
2. Calculate the average rate of consumption for each subinterval by taking the average of the values of R(t) at the endpoints of the subinterval. For example, for the first subinterval [0, 1.5], the average rate of consumption (R_avg1) is (400 + 200)/2 = 300 KW/hr.
3. Calculate the area of each trapezoid for each subinterval. The area of a trapezoid is given by the formula A = (base1 + base2)/2 * height, where the bases are the average rates of consumption for the subinterval and the height is the width of the subinterval. For the first subinterval, the area (A1) is (300 + 200)/2 * 1.5 = 375 KW.
4. Repeat steps 2 and 3 for each of the remaining subintervals.
5. Finally, add up the areas of all four trapezoids to estimate the total amount of electricity consumed between midnight and 6 a.m. In this case, it would be A1 + A2 + A3 + A4.
Using the given data, the calculations would be as follows:
Subinterval 1: [0, 1.5]
R_avg1 = (400 + 200)/2 = 300 KW/hr
A1 = (300 + 200)/2 * 1.5 = 375 KW
Subinterval 2: [1.5, 3]
R_avg2 = (200 + 350)/2 = 275 KW/hr
A2 = (275 + 350)/2 * 1.5 = 487.5 KW
Subinterval 3: [3, 4.5]
R_avg3 = (350 + 200)/2 = 275 KW/hr
A3 = (275 + 200)/2 * 1.5 = 412.5 KW
Subinterval 4: [4.5, 6]
R_avg4 = (200 + 150)/2 = 175 KW/hr
A4 = (175 + 150)/2 * 1.5 = 262.5 KW
Total estimated electricity consumed = A1 + A2 + A3 + A4 = 375 + 487.5 + 412.5 + 262.5 = 1537.5 KW
Therefore, the estimated total amount of electricity consumed between midnight and 6 a.m. is approximately 1537.5 KW.